Dr. Barrios-Faundez, Tomas

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dual-mixed approach suitable residual type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice

We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress–velocity formulation. The stabilized formulation is obtained by adding to the dual-mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations. We prove that for appropriate values of the stabilization parameters, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart–Thomas or Brezzi–Douglas–Marini elements and the velocity is approximated by continuous piecewise polynomials. Moreover, we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that support the theoretical results.